Möbius strip

A Möbius band Möbius loop or Möbius'sches band is a two-dimensional structure in the topology, which has only one edge and one surface. It is not orientable, that is, one can not distinguish between the top and bottom or between inside and outside.

It was described in 1858 independently by the Göttingen mathematician and physicist Johann Benedict Listing and Leipzig mathematician and astronomer August Ferdinand Möbius.

• 5.1 Analysis
• 5.2 topology
• 5.3 spinors

Description

A Möbius strip is easy to produce, by a long strip of paper to stick together in ring shape with both ends, but one end twisted by 180 ° from sticking together.

Such Möbius bands have a center line which can not take a circle - unless the tape is stretched locally. The mold, which can take such a tape unstretched, is fully described by the curve of the center line.

Möbius bands whose center line is a circle in the relaxed state, can not be made ​​from a straight two-dimensional piece of paper - they own along its circumference unevenly shaped sub-elements, from which they can be thought composed.

Möbius bands are chiral.

The Möbius band is so into himself, that if you start on one of the seemingly two sides to color the surface, has colored the end the whole object.

Another interesting effects arise when inscribing on the tape a center line parallel to the center line or two lines, and the band along the line (s) cuts open, so it appears to be halved, or into thirds. In the first case, ie when cutting along the center line, creating a double- twisted ( to 720 ° twisted in itself ) ring with two sides and two edges. In the second case, two objects arise: A Möbius band and a double twisted ring that hang together. This game can be continued with an arbitrary small division: " Quarters " you are band creates two double twisted tapes, which depend not only each other, but also once again wrapped around each other more frequently; " Fünftelt " to it, creates the same character with an additional Möbius strip that hangs in the two rings; " Sechstelt " to the band, you get two rings that wrap around twice and are double wrapped by another ring, the outer and the two inner rings are freely interchangeable; " Siebtelt " you turn it to come back for a Möbius band added that hangs in the three rings, etc. If n is the denominator of the fraction, in which one divides the band apparently, and n is even, so that n = 2r, we obtain r Rings; n is odd, n = 2r 1, then in addition looped a Möbius strip through the rings.

Mathematically, the Möbius strip is a non- orientable manifold. Another area that falls into this category, is the Klein bottle; one can decompose a Klein bottle into two parts so that from their results in two Möbius bands.

The mathematical symbol for infinity is sometimes misinterpreted as a Möbius strip.

In nature

• Charged particles were trapped in the magnetic field of the earth, can move on a Möbius strip
• The cyclic protein Kalata B1, active ingredient of the plant Oldenlandia. O. affinis, as a natural remedy for example, for the induction of labor, has a Möbius topology

In art and literature

Famous representations of the Möbius strip in the art, there are, for example, by MC Escher ( Mobius Band I and II, 1963 ) and more recently by Gideon Möbius Sherman. Even the Argentine feature film Moebius deals with the topic. In the literature, the Möbius strip is also discussed: the structure of John Barth's short story series " Lost in the Funhouse " (Eng. " Ambrose in Juxhaus " ) is based on the Infinity or principle of repetition (eg missing middle) of the Möbius strip. A Möbius strip the postmodern literature approaches ( "Frame - Tale" ) the book is included, reflects. It is inscribed with: "Once upon a time there was a story did began once upon a time ... ". This form of self-reference is typical of so-called strange loops. The poet Erich Fried refers in his poem " Topologik " on the Möbius strip: "I have taken me a Möbius heart that cuts into hopeless stripes. " Max Bill created from the 1930s, numerous sculptures that match the visual representations of the Möbius strip: eg ' Infinite Loop ' (1935 /37), ' continuity ' ( Lake Zurich, in 1947, destroyed 1948) or ' Infinite Loop ' (city garden food, to the Hohenzollern street; 1974). However, his sculpture continuity (1986 ) does not represent a Moebius strip, contrary to common opinion.

Also in existence since 1986 Necroscope series of novels by the English author Brian Lumley the Möbius band plays an important role. It is the symbol of some figures, but especially important for the main character Harry Keogh. He learned the skill of time travel with the help of the so-called Möbius continuum, which behaves similarly to the Möbius band.

Likewise, the Möbius strip in the Perry Rhodan series is discussed and here is the three-dimensional model description for the two sides of the n-dimensional universe ( Arresum and Paresum ).

In the manga series " Angel Sanctuary " is the fate of the high angel Alexiel and the constant rebirth of their souls into human bodies, where a cruel and bloody destiny is predetermined, compared with a Möbius Scheife.

In 2011, published in German language novel by Michel Houellebecq area map and a Möbius band is engraved on the grave stone of the fictional character Michel Houellebecq.

In 2011, the student robotics Aaron Hoover has a Möbius gear made ​​at the Berkeley University as a gadget using 3D printing.

The Möbius Chess is a variant of the Cylinder Chess, in which we add thinks the " connection" of the longitudinal sides a twist of the pitch.

In fashion, have also been designed Möbius scarves.

In the drama Solaris Stanislaw Lem by Bettina Bruinier and Katja Friedrich at the Munich National Theatre (2011 ) is a model of a car trafficked Möbius strip important part of staging (stage: Markus Karner ).

The Commerzbank logo shows a Möbius strip.

The East German avant-garde band AG. Violin dedicated to the Möbius band released a song on the 1989 album "Trick Beat".

In technology

Mechanics

• For belt drives, where it ensures uniform wear.

Electrical Engineering

• The circuitry analog of a Möbius band is a ring counter with an inversion ( Johnson counter ): a sequence of bits obtained after two rounds of the initial state, and therefore can be counted with n memory cells to 2n; Counting very fast successive pulses.
• A compact resonator with the resonance frequency at one-half of identical linear coils.
• An induction -free resistor that is also called Möbius resistance.

Physics

• The Stellarator is a type of a nuclear fusion reactor, wherein the plasma is brought to a Möbius shaped track by correspondingly shaped field coils.

Chemistry

• As a " node " molecules with special properties ( knotanes, chirality )

Nanotechnology

• As molecular motors.
• As graphene band (nano - graphite) with novel electronic properties, such as helical magnetism

In mathematics

Analysis

The Möbius strip can be drawn as a subset of the parameters using the following representation:

Where and. Thus a Mobius strip having a width of 1 is generated, the center line is located in the xy-plane and the center of circle of radius 1. The angle has its vertex in the center; while it changes, the variation results from the surface, which spans between the single edge. As is easy to see right in the picture, it is not a from a strip of paper to be manufactured Mobius Band - on the horizontal part resemble the partial elements symmetrical trapezoids.

Using cylindrical coordinates, an unlimited version of the Moebius strip is defined by the following equation:

Topology

The topology provides a mathematical way to make a Möbius strip by the opposing gluing together the ends of a strip of paper. There, a Mobius band is defined as the quotient of the square space, where two opposite sides are identified by the equivalence relation for each other. The diagram illustrates this.

Spinors

One can interpret the edge of the Möbius strip as a spinor: The group is parameterized by. The spinor can be seen as subset

Interpret; this is exactly the edge of the Moebius strip

New insights into the mathematical description of a Möbius bands were in 2007 by the scientists EL Starostin and G.H.M. van der Heijden published. Particular, they have the shape mathematically calculated, which strives to assume a product manufactured from a band Möbius band by itself, so as to assume the lowest energy state.